Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy

We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class $C_{loc}^{1,\alpha }$, for some constant $\alpha \in (0,1)$. In addition, under suitable conditions on $\sigma$, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.